Stability of monotone travelling waves for competition-diffusion equations

  • Yukio Kan-On
  • Qing Fang
Article

Abstract

We consider the Lotka-Volterra competition model with diffusion inR, and establish a theorem on the asymptotic stability of monotone travelling waves relative to the space of bounded uniformly continuous functions with the supremum norm. In consideration of the result of Alexander et al. [1] and Derndinger [4], we shall arrive at a study of the non-negative eigenvalues of the linearized operator around the travelling wave.

Key words

competition-diffusion system travelling wave stability 

References

  1. [1]
    J. Alexander, R. Gardner and C. Jones, A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math.,410 (1990), 167–212.MATHMathSciNetGoogle Scholar
  2. [2]
    P. W. Bates and C. K. R. T. Jones, Invariant, manifolds for semilinear partial differential equations. Dynamics Reported Vol.2 (eds. U. Kirchgraber et al.), Wiley Chichester, 1989, 1–38.Google Scholar
  3. [3]
    E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.MATHGoogle Scholar
  4. [4]
    R. Derndinger, Über das Spektrum positiver Generatoren. Math. Z.,172 (1980), 281–293.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach. J. Differential Equations,44 (1982), 343–364.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Gardner and C. K. R. T. Jones, Stability of travelling wave solutions of diffusive predatorprey systems. Trans. Amer. Math. Soc.,327 (1991), 465–524.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math.840, Springer-Verlag, New York-Berlin-Tokyo, 1981.MATHGoogle Scholar
  8. [8]
    Y. Hosono, Singular perturbation analysis of travelling waves of diffusive Lotka-Volterra competition models. Numerical and Applied Mathematics Part II (Paris, 1988) Baltzer, Basel, 1989, 687–692.Google Scholar
  9. [9]
    Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. Preprint.Google Scholar
  10. [10]
    Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations. Hiroshima Math. J.23 (1993), 193–221.MATHMathSciNetGoogle Scholar
  11. [11]
    M. Mimura and P. C. Fife, A 3-component, system of competition and diffusion. Hiroshima Math. J.,16 (1986), 189–207.MATHMathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 1996

Authors and Affiliations

  • Yukio Kan-On
    • 1
  • Qing Fang
    • 2
  1. 1.Department of Mathematics, Faculty of EducationEhime UniversityMatsuyamaJapan
  2. 2.Department of Mathematics, Faculty of ScienceEhime UniversityMatsuyamaJapan

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